高波数Helmholtz方程的有限元方法和连续内罚有限元方法

doi:10.12286/jssx.2018.2.191

计算数学 ›› 2018, Vol. 40 ›› Issue (2): 191-213.DOI: 10.12286/jssx.2018.2.191

高波数Helmholtz方程的有限元方法和连续内罚有限元方法

武海军

南京大学数学系, 南京 210093

收稿日期:2017-08-31 出版日期:2018-06-15 发布日期:2018-05-15
基金资助:
国家自然科学基金（11525103，91630309，11621101）资助.

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武海军. 高波数Helmholtz方程的有限元方法和连续内罚有限元方法[J]. 计算数学, 2018, 40(2): 191-213.

Wu Haijun. FEM AND CIP-FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER[J]. Mathematica Numerica Sinica, 2018, 40(2): 191-213.

FEM AND CIP-FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER

Wu Haijun

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received:2017-08-31 Online:2018-06-15 Published:2018-05-15

本文介绍高波数Helmholtz方程的有限元方法和连续内罚有限元方法.将以线性元情形为例，给出方法的明显依赖于波数k的预渐近稳定性和误差分析.我们将介绍三种证明方法.我们还讨论了内罚有限元方法的罚参数的选取以显著减少方法的污染误差.最后还给出数值例子验证理论结果.

关键词： Helmholtz方程, 高波数, 内罚有限元方法, 预渐近误差估计

Finite element methods (FEM) and continuous interior penalty finite element methods (CIP-FEM) are considered for the Helmholtz equation with high wave number. Preasymptotic stability and error analyses with explicit dependence on the wave number k are provided for the linear versions of the methods. Three approaches for the analyses will be introduced. In order to reduce greatly the pollution error of the methods, the choice of the penalty parameters for the CIP-FEM will be discussed. Numerical examples are given to verify the theoretical results.

Key words: Helmholtz equation, high wave number, FEM, CIP-FEM, Preasymptotic error estimates

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高波数Helmholtz方程的有限元方法和连续内罚有限元方法

FEM AND CIP-FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER

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